History of Topics in Special Relativity/Lorentz transformation (conformal)
History of Topics in Special Relativity: History of Lorentz transformation ( )  

Lorentz transformation via sphere transformation[edit  edit source]
If one only requires the invariance of the light cone represented by the differential equation , which is the same as asking for the most general transformation that changes spheres into spheres, the Lorentz group can be extended by adding dilations represented by the factor λ. The result is the group Con(1,p) of spacetime w:conformal transformations in terms of w:special conformal transformations and inversions producing the relation
 .
One can switch between two representations of this group by using an imaginary sphere radius coordinate x_{0}=iR with the interval related to conformal transformations, or by using a real radius coordinate x_{0}=R with the interval related to Lie's (1871) sphere transformation (or w:spherical wave transformations) in terms of w:contact transformations preserving circles and spheres. It was shown by Bateman & Cunningham (1909–1910), that the group Con(1,3) is the most general one leaving invariant the equations of Maxwell's electrodynamics.
It turns out that Con(1,3) is isomorphic to the w:special orthogonal group SO(2,4), and contains the Lorentz group SO(1,3) as a subgroup by setting λ=1. More generally, Con(q,p) is isomorphic to SO(q+1,p+1) and contains SO(q,p) as subgroup.^{[1]} This implies that Con(0,p) is isomorphic to the Lorentz group of arbitrary dimensions SO(1,p+1). Consequently, the conformal group in the plane Con(0,2) – known as the group of w:Möbius transformations – is isomorphic to the Lorentz group SO(1,3).^{[2]}^{[3]} This can be seen using tetracyclical coordinates satisfying the form , which were discussed by Pockels (1891), Klein (1893), Bôcher (1894). The relation between Con(1,3) and the Lorentz group was noted by Bateman & Cunningham (1909–1910) and others.
A subgroup of Lie's group of sphere transformations is the Laguerre group (or group of transformations by reciprocal directions) dealing with oriented spheres, planes and lines, which was already implicit in the work of Ribaucour (1870), Lie (1871), Darboux (1873). It's generated by the Laguerre inversion introduced by Laguerre (1882) and discussed by Darboux (1887) and Smith (1900) leaving invariant with R as radius, thus the Laguerre group is isomorphic to the Lorentz group. A similar concept was studied by Scheffers (1899) in terms of contact transformations. Stephanos (1883) argued that Lie's geometry of oriented spheres in terms of contact transformations, as well as the special case of the transformations of oriented planes into each other (such as by Laguerre), provides a geometrical interpretation of Hamilton's w:biquaternions. The w:group isomorphism between the Laguerre group and Lorentz group was pointed out by Bateman (1910), Cartan (1912, 1915/55), Poincaré (1912/21) and others.^{[4]}^{[5]} The Laguerre inversion was written in the following ways:

(
)
or

(
)
A special case of Laguerre inversions (1, 2) with was already given by Bonnet (1856). Formula (1) was given by Laguerre (1882) and Darboux (1887), formula (2) by Smith (1900). Laguerre transformations in trigonometric form were given by Scheffers (1899).
Formulas (1, 2) become ordinary Lorentz transformations by changing the sign of and setting . Furthermore, using together with and gives:

(
)
Formula (3) was already used by Darboux (1873) as a sphere transformation, and in 1881 he showed that it can also be used to perform Laguerre transformations of planes. In special relativity, it turns out that formula (3) describes the aberration of light, see velocity addition and aberration.
Historical notation[edit  edit source]
Bonnet (1856)[edit  edit source]
w:Pierre Ossian Bonnet (1856) defined a reciprocal transformation preserving lines of curvatures. He noted that his transformation implies the following relation between curvature radii and ordinates of the respective curvature centers:^{[M 1]}
 where
Bonnet's transformation produces and represents a special case of Laguerre inversion (or Lorentz transformation) (1, 2) with . Lie (1871), Darboux (1887) and Smith (1900) all noticed that Bonnet's transformation is a special case of Lie's and Laguerre's transformations.
Ribaucour (1870)[edit  edit source]
w:Albert Ribaucour (1870),^{[M 2]} defined what was later called “Ribaucour transformations” preserving lines of curvature:
 p. 330: If circles are normal to three surfaces, they are normal to a family of surfaces belonging to a triply orthogonal system. This results in a class of orthogonal triple systems which I will propose to call cyclic systems, intimately linked to the deformation of surfaces. Given a surface (A), we can propose to seek all the cyclic systems which derive from it; the of this surface being put in the form [...]
 p. 332: If spheres have their contact chords normal to surfaces, the circles passing through the centers of these spheres and their points of contact with their enveloping surfaces are normal to an infinity of surfaces forming part of a cyclic system. [...] If surfaces are part of an orthogonal system, the osculating circles of their orthogonal trajectories corresponding to all the points of one of these surfaces are normal to a family of surfaces belonging to a cyclic system. [...] I will point out the simple case where (A) is a plane, a case which leads to a general transformation of the surfaces with correspondence of the lines of curvature [...].
Referring to p. 332 of Ribaucour's paper, Darboux (1887) and Bateman (1910) argued that Ribaucour anticipated both Lie (1871) and Laguerre (1880) in formulating the “transformation by reciprocal directions”, which in Darboux's representation led to algebraic expressions identical to Laguerre inversion (or Lorentz transformation) (1) and (3).
Lie (1871)[edit  edit source]
In several papers between 1847 and 1850 it was shown by w:Joseph Liouville^{[M 3]} that the relation λ(δx^{2}+δy^{2}+δz^{2}) is invariant under the group of w:conformal transformations generated by w:inversions transforming spheres into spheres, which can be related w:special conformal transformations or w:Möbius transformations. (The conformal nature of the linear fractional transformation of a complex variable was already discussed by Euler (1777)).^{[M 4]}^{[6]}
Liouville's theorem was extended to all dimensions by w:Sophus Lie (1871a).^{[M 5]}^{[7]} In addition, Lie described a manifold whose elements can be represented by spheres, where the last coordinate y_{n+1} can be related to an imaginary radius by iy_{n+1}:^{[M 5]}
If the second equation is satisfied, two spheres y′ and y″ are in contact. Lie then defined the correspondence between w:contact transformations in R_{n} and conformal point transformations in R_{n+1}: The sphere of space R_{n} consists of n+1 parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space R_{n}, it follows that R_{n} now corresponds to R_{n+1}. Therefore, any transformation (to which he counted E:orthogonal transformations and inversions) leaving invariant the condition of contact between spheres in R_{n}, corresponds to the conformal transformation of points in R_{n+1}. He pointed out that conformal point transformations consist of motions (such as w:rigid transformations and orthogonal transformations), similarity transformations, and inversions.^{[M 6]}
As shown by Bateman and Cunningham (1909), the spacetime conformal group Con(1,3) of "w:spherical wave transformations" corresponds to the transformations of Lie's sphere geometry in which the radius indicates the fourth coordinate, while the Lorentz group SO(1,3) is a subgroup of Con(1,3). It's also known that the Möbius group and Laguerre group, which are both isomorphic to the Lorentz group, are subgroups of Lie's sphere transformations group.
In the same paper, Lie also mentioned the “well known fact” that “parallel transformations” (dilatations having the property of transforming planes to parallel planes) preserve lines of curvature, and he alluded to Bonnet's (1856) transformation as an example.^{[M 7]} Generally, all of the discussed transformations that preserve lines of curvature are either inversions or parallel transformations.^{[M 8]} In a footnote he specifically remarked that line transformations under which "(const=0)" remains unchanged, give all transformations of R by which surfaces of common spherical image pass into other such surfaces, and that the new spherical image emerges from the former by a conformal point transformation of the imagesphere, and that Bonnet's (1856) transformation belongs here.^{[M 9]}
Lie himself (1884)^{[M 10]} pointed out that his remarks indicate the same transformation group treated in more recent works of Laguerre (188082) and Stephanos (1882). Consequently, Smith (1900) credits Lie as being the first one to allude to the existence of the (extended) Laguerre group, transforming spheres into spheres and planes into planes. On the other hand, Darboux (1887) pointed out that the transformation by reciprocal directions was already anticipated by Ribaucour (1870) even before Lie.
Klein, Pockels, Bôcher (187191)[edit  edit source]
In relation to line geometry, w:Felix Klein (1871/72)^{[M 11]} used coordinates satisfying the condition . They were introduced in 1868 (belatedly published in 1873) by w:Gaston Darboux^{[M 12]} as a system of five coordinates in R_{3} (later called "pentaspherical" coordinates) in which the last coordinate is imaginary. w:Sophus Lie (1871)^{[M 13]} more generally used n+2 coordinates in R_{n} (later called "polyspherical" coordinates) satisfying in which the last coordinate is imaginary, as a means to discuss conformal transformations generated by inversions. These simultaneous publications can be explained by the fact that Darboux, Lie, and Klein corresponded with each other by letter.
When the last coordinate is defined as real, the corresponding polyspherical coordinates satisfy the form of a sphere. Initiated by lectures of Klein between 1889–1890, his student w:Friedrich Carl Alwin Pockels (1891) used such real coordinates, emphasizing that all of these coordinate systems remain invariant under conformal transformations generated by inversions:^{[M 14]}
Special cases were described by Klein (1893):^{[M 15]}
 (pentaspherical).
 (tetracyclical).
Both systems were also described by w:Maxime Bôcher (1894) in an expanded version of a thesis supervised by Klein.^{[M 16]}
Polyspherical coordinates indicate that the conformal group Con(0,p) is isomorphic to the Lorentz group SO(1,p+1).^{[8]} For instance, Con(0,2) – known as Möbius group – is related to tetracyclical coordinates satisfying , which is nothing other than the Lorentz interval invariant under the Lorentz group SO(1,3).
Darboux (187387)[edit  edit source]
In 1873, w:Gaston Darboux stated the following proposition:^{[M 17]}
 Given a surface , we add a fixed sphere to it, and we construct all spheres tangent to the surface and intersecting at a constant angle . Through the intersection of each of these spheres and new spheres pass intersecting at a constant angle . These new spheres envelop a surface , corresponding point by point to with conservation of lines of curvature. The corresponding points on the two surfaces are on circles normal both to the two surfaces and to the sphere .
which he generalized by making a second proposition:^{[M 18]}
 Consider a surface , envelope of a series of variable spheres intersecting under any angles the sphere . At each of the spheres intersecting at an angle I call we match a sphere passing through the intersection of and from , and intersecting at an angle determined by equation
 Then the new spheres envelop a surface which corresponds point by point at with curvature lines preserved. If we subject the spheres tangent to to cut under a constant angle, will be constant; it will be the same for , by virtue of the previous equation, and we find the theorem given above. »
This is equivalent to Laguerre transformation (or Lorentz transformation) (3) with .
In 1881 he quoted his above propositions, gave priority to the first one to Ribaucour (1870), and then showed that Laguerre's transformation of reciprocal directions is included as well:^{[M 19]}
 This proposal gave a new means of realizing a mode of transformation of surfaces with preservation of the lines of curvature, to which Ribaucour had devoted a few lines in a Communication made to the Academy in 1870 sur la deformation des surfaces.
 [..] Suppose, in particular, that the sphere reduces to a plane . Then to any plane will correspond a plane passing through the intersection of and , and the angles that the planes , make with will be linked by relation (1). It is not difficult to recognize, in this transformation from one plane to another, that which has recently been studied by Laguerre under the name of transformation by reciprocal directions. We see that it is included in the transformation of spheres which is defined by our second proposition. I have recalled these results only to arrive at the proposition which is the main object of this Communication. I will show, in accordance with a general theorem of Lie, that the transformation first proposed by Ribaucour boils down to dilatations (transition from a surface to the parallel surface) and to transformations by reciprocal vector rays.
He went on to rewrite his 1873 equation as:^{[M 20]}
This is equivalent to Laguerre transformation (or Lorentz transformation) (3) with .
In 1887, Darboux gave a much more detailed account. For instance, he rederived and extended the transformation of oriented halflines given by Laguerre (1882) using coordinates x,y,z,R:^{[M 21]}
 or
He went on to derive expressions and theorems similar to those given by him in 1873, and added that Bonnet's (1856) transformation is a special case.^{[M 22]}
This is equivalent to Laguerre inversion (or Lorentz transformation) (1).
Regarding the history of such transformations (before Laguerre's research) he wrote:^{[M 23]}
 In the memoir already quoted, inserted in volume V of Mathematische Annalen, Lie has made known all the contact transformations which preserve the lines of curvature; he even pointed out (p. 186) the particular case of transformation by reciprocal directions; but this transformation had already been given in different works by Ribaucour. See, in particular, Ribaucour's note sur la deformation des surfaces (Comptes rendus, t. LXX, p. 332, 1870). In a different form, it was the subject of the author's studies published in Notes V and IX of Mémoire sur une classe remarquable de courbes et de surfaces algébriques, 1873.
Laguerre (188082)[edit  edit source]
A systematic formulation of a geometry of orientation was given by w:Edmond Laguerre (1880), including geometric transformations of oriented planes into oriented planes and oriented spheres into oriented spheres, which he called "w:transformation by reciprocal directions".^{[M 24]} Besides the focus on the transformation of planes, a distinguishing feature to previous authors was the employment of the concept of orientation (i.e. attributing a certain sign to lines and radii) which became an indispensable tool in Lie sphere geometry and Laguerre geometry.
Laguerre's transformations form a group (Laguerre group) which is isomorphic to the Lorentz group, and forms a subgroup of Lie's (1871) contact transformations of spheres.
In 1882 he developed the “transformation of oriented halflines” which was later called "Laguerre inversion", using the following algebraic formulation (R being the radius and D the distance of its center to the axis):^{[M 25]}
This is equivalent to Laguerre inversion (or Lorentz transformation) (1). The Laguerre inversions are generators of the Laguerre group.
Stephanos (1883)[edit  edit source]
w:Cyparissos Stephanos (1883)^{[M 26]} showed that Hamilton's biquaternion a_{0}+a_{1}ι_{1}+a_{2}ι_{2}+a_{3}ι_{3} can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector a_{1}ι_{1}+a_{2}ι_{2}+a_{3}ι_{3} as its center and the scalar as its radius. Its norm is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N(Q_{1}Q_{2}) (the corresponding spheres are in contact with N(Q_{1}Q_{2})=0) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a w:homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:^{[M 27]}^{[9]}^{[10]}
 (or ).
Stephanos pointed out that the special case A=0 denotes transformations of oriented planes (see Laguerre (1882)).
The Lorentz group SO(1,3) is a subgroup of the conformal group Con(1,3) in terms of Lie's (1871) transformations of oriented spheres in which the radius indicates the fourth coordinate. The Lorentz group is isomorphic to the group of Laguerre's (1880) transformation of oriented planes.
Scheffers (1899)[edit  edit source]
w:Georg Scheffers (1899) synthetically determined all finite w:contact transformations preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines (compare with Laguerre inversion by Laguerre (1882) and Darboux (1887)):^{[M 28]}
This is equivalent to Laguerre transformation (or Lorentz transformation) (1) by the identity .
Smith (1900)[edit  edit source]
w:Percey F. Smith (1900) followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:^{[M 29]}
He added that Bonnet's (1856) transformation is a special case with , and he also gave credit to Lie (1871) for defining the corresponding "group of the geometry of reciprocal directions".
This is equivalent to Laguerre inversion (or Lorentz transformation) (2).
Bateman and Cunningham (1909–1910)[edit  edit source]
In line with Lie's (1871) research on the relation between sphere transformations with an imaginary radius coordinate and 4D conformal transformations, it was pointed out by w:Harry Bateman and w:Ebenezer Cunningham (1909–1910), that by setting u=ict as the imaginary fourth coordinates one can produce spacetime conformal transformations. Not only the quadratic form , but also w:Maxwells equations are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were called w:spherical wave transformations by Bateman.^{[R 1]}^{[R 2]} However, this covariance is restricted to certain areas such as electrodynamics, whereas the totality of natural laws in inertial frames is covariant under the w:Lorentz group.^{[R 3]} In particular, by setting λ=1 the Lorentz group SO(1,3) can be seen as a 10parameter subgroup of the 15parameter spacetime conformal group Con(1,3).
Bateman (1910/12)^{[11]} also alluded to the identity between the Laguerre inversion and the Lorentz transformations. In general, the isomorphism between the Laguerre group and the Lorentz group was pointed out by w:Élie Cartan (1912, 1915/55),^{[5]}^{[R 4]} w:Henri Poincaré (1912/21)^{[R 5]} and others.
References[edit  edit source]
Historical mathematical sources[edit  edit source]
 ↑ Bonnet (1856), p. 487
 ↑ Ribaucour (1870). pp. 330333
 ↑ Liouville (1847)
 ↑ Euler (1777), p. 140
 ↑ ^{5.0} ^{5.1} Lie (1871), pp. 199–209
 ↑ Lie (1871/72), first footnote on p. 186
 ↑ Lie (1871/72), p. 184
 ↑ Lie (1871/72), p. 186
 ↑ Lie (1871/72), second footnote on p. 186
 ↑ Lie (1884), footnote on p. 541
 ↑ Klein (1871/72), p. 268
 ↑ Darboux (1873), p. 137
 ↑ Lie (1871), p. 208
 ↑ Pockels (1891), pp. 197–206
 ↑ Klein (1893c), pp. 200ff (pentaspherical), pp. 373ff (tetracyclical)
 ↑ Bôcher (1894), pp. 30–34, 40–43
 ↑ Darboux (1873), pp. 254255.
 ↑ Darboux (1873), footnote on p. 255.
 ↑ Darboux (1881), p. 286f.
 ↑ Darboux (1881), footnote on p. 287
 ↑ Darboux (1887), p. 254
 ↑ Darboux (1887), p. 256
 ↑ Darboux (1887), footnote on p. 259
 ↑ Laguerre (1880)
 ↑ Laguerre (1882), pp. 550–551.
 ↑ Stephanos (1883), p. 590ff
 ↑ Stephanos (1883), p. 592
 ↑ Scheffers (1899), p. 158
 ↑ Smith (1900), p. 159
 Bôcher, M. (1894). Über die Reihenentwicklungen der Potentialtheorie. Leipzig: Macmillan. https://archive.org/details/acv1533.0001.001.umich.edu.
 Bonnet, P.O. (1856). "Note sur un genre particulier de surfaces réciproques". Compt. Rend. 42: 485487. https://gallica.bnf.fr/ark:/12148/bpt6k2999t/f489.item.
 Darboux, G. (1873). Sur une classe remarquable de courbes et de surfaces algébriques et sur la théorie des imaginaires. GauthierVillars. https://archive.org/details/suruneclasserema00darb.
 Darboux, G. (1881). "Sur les modes de transformation qui conservent les lignes de courbure". Compt. Rend. 92: 286289.
 Darboux, G. (1887). Leçons sur la théorie générale des surfaces. Première partie. Paris: GauthierVillars. pp. 254–256. https://archive.org/details/leconsurlagendes01darbrich.
 Euler, L. (1777). "De proiectione geographica superficiei sphaericae". Acta Academiae Scientiarum Imperialis Petropolitanae: 133–142. http://eulerarchive.maa.org/pages/E491.html.
 Klein, F. (1872) [1871]. "Ueber Liniengeometrie und metrische Geometrie". Mathematische Annalen 5: 257–277. doi:10.1007/bf01444841. http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002241854.
 Klein, F. (1893c). Schilling, Fr.. ed. Einleitung in die höhere Geometrie I, Vorlesung gehalten während des Wintersemesters 1892–93. Göttingen. https://archive.org/details/eininhohere01kleirich.
 Laguerre, Edmond (1880). "Sur la géométrie de direction". Bulletin de la Société Mathématique de France 8: 196208. doi:10.24033/bsmf.207.
 Laguerre, Edmond (1882). "Transformations par semidroites réciproques". Nouvelles annales de mathématiques 1: 542–556.
 Lie, S. (1871a). "Ueber diejenige Theorie eines Raumes mit beliebig vielen Dimensionen, die der KrümmungsTheorie des gewöhnlichen Raumes entspricht". Göttinger Nachrichten: 191–209. https://books.google.com/books?id=mH41bQAAMAAJ&pg=PA191.
 Lie, S. (1872) [1871]. "Ueber Complexe, insbesondere Linien und KugelComplexe, mit Anwendung auf die Theorie partieller Differentialgleichungen". Mathematische Annalen 5: 145–256. doi:10.1007/bf01446331. https://books.google.com/books?id=TdZSAAAAcAAJ&pg=PA145. English translation by David Delphenich: On complexes – in particular, line and sphere complexes – with applications to the theory of partial differential equations
 Lie, S. (1884). "Ueber Differentialinvarianten". Math. Ann. 24: 537–578. http://resolver.sub.unigoettingen.de/purl?PPN235181684_0024.
 Liouville, Joseph (1850). "Théorème sur l'équation dx²+dy²+dz²=λ(dα²+dβ²+dγ²)". Journal de Mathématiques Pures et Appliquées 15: 103. https://books.google.com/books?id=wlZOAAAAMAAJ&pg=PA103.
 Pockels, F. (1891). Über die partielle Differentialgleichung Δu+k²u=0 und deren Auftreten in der mathematischen Physik. Leipzig: Teubner. https://archive.org/details/berdiepartiell02pockuoft.
 Ribaucour, Albert (1870). "Sur la déformation des surfaces". Comptes Rendus 70: 330–333.
 Scheffers, G. (1899). "Synthetische Bestimmung aller Berührungstransformationen der Kreise in der Ebene". Leipziger Math.Phys. Berichte 51: 145–160. https://hdl.handle.net/2027/hvd.32044092889328.
 Smith, Percey F. (1900). "On a Transformation of Laguerre". Annals of Mathematics 1 (1/4): 153–172. doi:10.2307/1967282. https://archive.org/details/jstor1967282.
 Stephanos, C. (1883). "Sur la théorie des quaternions". Mathematische Annalen 7 (4): 589–592. doi:10.1007/bf01443267. http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002247798.
Historical relativity sources[edit  edit source]
 Bateman, H. (1910) [1909]. "The Transformation of the Electrodynamical Equations". Proceedings of the London Mathematical Society 8: 223264. http://hdl.handle.net/2027/inu.30000021006535.
 The Transformation of the Electrodynamical Equations on English Wikisource
 Bateman, Harry (1912) [1910]. "Some geometrical theorems connected with Laplace's equation and the equation of wave motion". American Journal of Mathematics 34 (3): 325–360. doi:10.2307/2370223. https://archive.org/details/jstor2370223.
 Cartan, Élie (1912). "Sur les groupes de transformation de contact et la Cinématique nouvelle". Société de Mathématique the France – Comptes Rendus des Séances: 23. https://archive.org/stream/bulletinsocit40soci#page/422/mode/2up.
 Cunningham, Ebenezer (1910) [1909]. "The principle of Relativity in Electrodynamics and an Extension Thereof". Proceedings of the London Mathematical Society 8: 77–98. doi:10.1112/plms/s28.1.77.
 The principle of Relativity in Electrodynamics and an Extension Thereof on English Wikisource
 Klein, Felix (1921) [1910]. Über die geometrischen Grundlagen der Lorentzgruppe. 1. 533–552. doi:10.1007/9783642519604_31. ISBN 9783642518980.
 Über die geometrischen Grundlagen der Lorentzgruppe on German Wikisource
 Poincaré, Henri (1921) [1912]. "Rapport sur les travaux de M. Cartan (fait à la Faculté des sciences de l'Université de Paris)". Acta Mathematica 38 (1): 137–145. doi:10.1007/bf02392064. https://archive.org/stream/actamathematica38upps#page/n153/mode/2up. Written by Poincaré in 1912, printed in Acta Mathematica in 1914 though belatedly published in 1921.
Secondary sources[edit  edit source]
 ↑ Schottenloher (2008), section 2.2
 ↑ Kastrup (2008), section 2.4.1
 ↑ Schottenloher (2008), section 2.3
 ↑ Coolidge (1916), p. 370
 ↑ ^{5.0} ^{5.1} Cartan & Fano (1915/55), sections 14–15
 ↑ Kastrup (2008), section 2.1
 ↑ Kastrup (2008), section 2.3
 ↑ Kastrup (2008), p. 22
 ↑ Cartan & Study (1908), p. 460
 ↑ Rothe (1916), p. 1399
 ↑ Bateman (1910/12), pp. 358–359
 Cartan, É.; Study, E. (1908). "Nombres complexes". Encyclopédie des Sciences Mathématiques Pures et Appliquées 1.1: 328–468. http://gallica.bnf.fr/ark:/12148/bpt6k2440f/f173.image.
 Cartan, É.; Fano, G. (1955). "La théorie des groupes continus et la géométrie". Encyclopédie des Sciences Mathématiques Pures et Appliquées 3.1: 39–43. http://gallica.bnf.fr/ark:/12148/bpt6k29100t/f194.image. (Only pages 1–21 were published in 1915, the entire article including pp. 39–43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the Encyclopédie in 1991.)
 Coolidge, Julian (1916). A treatise on the circle and the sphere. Oxford: Clarendon Press.
 Kastrup, H. A. (2008). "On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics". Annalen der Physik 520 (9–10): 631–690. doi:10.1002/andp.200810324.
 Klein, Felix; Blaschke, Wilhelm (1926). Vorlesungen über höhere Geometrie. Berlin: Springer. http://resolver.sub.unigoettingen.de/purl?PPN373601816.
 Rothe, H. (1916). "Systeme geometrischer Analyse". Encyclopädie der Mathematischen Wissenschaften 3.1.1: 1282–1425. http://resolver.sub.unigoettingen.de/purl?PPN360609767.
 Schottenloher, M. (2008). A Mathematical Introduction to Conformal Field Theory. Springer. ISBN 9783540706908.